## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 18, Number 4 (2008), 1548-1568.

### Central limit theorem for a many-server queue with random service rates

#### Abstract

Given a random variable *N* with values in ℕ, and *N* i.i.d. positive random variables {*μ*_{k}}, we consider a queue with renewal arrivals and *N* exponential servers, where server *k* serves at rate *μ*_{k}, under two work conserving routing schemes. In the first, the service rates {*μ*_{k}} need not be known to the router, and each customer to arrive at a time when some servers are idle is routed to the server that has been idle for the longest time (or otherwise it is queued). In the second, the service rates are known to the router, and a customer that arrives to find idle servers is routed to the one whose service rate is greatest. In the many-server heavy traffic regime of Halfin and Whitt, the process that represents the number of customers in the system is shown to converge to a one-dimensional diffusion with a random drift coefficient, where the law of the drift depends on the routing scheme. A related result is also provided for nonrandom environments.

#### Article information

**Source**

Ann. Appl. Probab., Volume 18, Number 4 (2008), 1548-1568.

**Dates**

First available in Project Euclid: 21 July 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1216677131

**Digital Object Identifier**

doi:10.1214/07-AAP497

**Mathematical Reviews number (MathSciNet)**

MR2434180

**Zentralblatt MATH identifier**

1145.60047

**Subjects**

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60F05: Central limit and other weak theorems 60K37: Processes in random environments 90B22: Queues and service [See also 60K25, 68M20] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

**Keywords**

Central limit theorem many-server queue random environment Halfin–Whitt regime heavy traffic routing policies fairness sample-path Little’s law

#### Citation

Atar, Rami. Central limit theorem for a many-server queue with random service rates. Ann. Appl. Probab. 18 (2008), no. 4, 1548--1568. doi:10.1214/07-AAP497. https://projecteuclid.org/euclid.aoap/1216677131